Picard's Great Theorem

Every nonconstant entire function attains every complex value with at most one exception (Henrici 1988, p. 216; Apostol 1997). Furthermore, every analytic function assumes every complex value, with possibly one exception, infinitely often in any neighborhood of an essential singularity.

REFERENCES:

Apostol, T. M. "Application to Picard's Theorem." §2.9 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 43-44, 1997.

Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, 1997.

Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, 1988.

Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, 1968.

Krantz, S. G. "Picard's Great Theorem." §10.5.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 140, 1999.

Narasimhan, R. and Nievergelt, Y. Complex Analysis in One Variable. Boston: Birkhäuser, 2001.

Remmert, R. Funktionentheorie 1. Berlin: Springer-Verlag, 1992.

Remmert, R. Funktionentheorie 2. Berlin: Springer-Verlag, 1992.

Trott, M. "Elementary Transcendental Functions." The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 166, 2004. http://www.mathematicaguidebooks.org/.